Some mathematical comments about the analytic hierarchy process: Part I - theoretical analysis

The Analytic Hierarchy Process (AHP) is a decision making method, which has as its greatest criticism the rank reversal effect. Here a new mathematical analysis of this method is performed, and three new results are highlighted. First, the method is formulated as a linear system of equations, where it is possible to assign a geometric interpretation, determine the number of possible solutions, and perform an sensitivity analysis based on the condition number of the matrix. Second, the causes of rank reversal can be encompassed by two mathematical aspects related to the properties of the matrix of AHP: high condition number and deficient rank. When the matrix is deficient rank, it is possible to obtain a condensed formulation of the AHP with a new full rank matrix. This guarantees greater stability to the method. Third, some mathematical results can be used as a robustness test for the matrix of AHP.